APMO 1997 Problem 5

Suppose that $n$ people $A_1$ , $A_2$ , $\ldots$ , $A_n$ , ( $n \geq 3$ ) are seated in a circle and that $A_i$ has $a_i$ objects such that $$a_1 + a_2 + \cdots + a_n = nN$$ where $N$ is a positive integer. In order that each person has the same number of objects, each person $A_i$ is to give or to receive a certain number of objects to or from its two neighbours $A_{i-1}$ and $A_{i+1}$ . (Here $A_{n+1}$ means $A_1$ and $A_n$ means $A_0$ .) How should this redistribution be performed so that the total number of objects transferred is minimum?